Direct sum decomposition.
if every can be expressed as a sum
for some and .
then we say that and form a direct sum decomposition of and write
In such circumstances, we also say that and are complementary subspaces, and also say that is an algebraic complement of .
Let be as above, and suppose that is finite-dimensional. The subspaces and are complementary if and only if for every basis of and every basis of , the combined list
is a basis of .
Specializing somewhat, suppose that the ground field is either the real or complex numbers, and that is either an inner product space or a unitary space, i.e. comes equipped with a positive-definite inner product
In such circumstances, for every subspace we define the orthogonal complement of , denoted by to be the subspace
Suppose that is finite-dimensional and a subspace. Then, and its orthogonal complement determine a direct sum decomposition of .
|Date of creation||2013-03-22 12:52:16|
|Last modified on||2013-03-22 12:52:16|
|Last modified by||rmilson (146)|